Basins of attraction for cascading maps

We study a finite uni-directional array of "cascading" or "threshold coupled" chaotic maps. Such systems have been proposed for use in nonlinear computing and have been applied to classification problems in bioinformatics. We describe some of the attractors for such systems and prove general results about their basins of attraction. In particular, we show that the basins of attraction have infinitely many path components. We show that these components always accumulate at the corners of the domain of the system. For all threshold parameters above a certain value, we show that they accumulate at a Cantor set in the interior of the domain. For certain ranges of the threshold, we prove that the system has many attractors.Comment: 15 pages, 9 figures. To appear in International Journal of Bifurcations and Chao

A stepwise pathway for biogenesis of 24-nt secondary siRNAs and spreading of DNA methylation

We used a transgene system to study spreading of RNA-directed DNA methylation (RdDM) during transcriptional gene silencing in Arabidopsis thaliana. Forward and reverse genetics approaches using this system delineated a stepwise pathway for the biogenesis of secondary siRNAs and unidirectional spreading of methylation from an upstream enhancer element into downstream sequences. Trans-acting, hairpin-derived primary siRNAs induce primary RdDM, independently of an enhancer-associated ‘nascent' RNA, at the target enhancer region. Primary RdDM is a key step in the pathway because it attracts the secondary siRNA-generating machinery, including RNA polymerase IV, RNA-dependent RNA polymerase2 and Dicer-like3 (DCL3). These factors act in a turnover pathway involving a nascent RNA, which normally accumulates stably in non-silenced plants, to produce cis-acting secondary siRNAs that induce methylation in the downstream region. The identification of DCL3 in a forward genetic screen for silencing-defective mutants demonstrated a strict requirement for 24-nt siRNAs to direct methylation. A similar stepwise process for spreading of DNA methylation may occur in mammalian genomes, which are extensively transcribed in upstream regulatory regions